Differential geometry and topology

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In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It arises naturally from the study of the theory of differential equations. Differential geometry is the study of geometry using differential calculus (cf. integral geometry). These fields are adjacent, and have many applications in physics, notably in the theory of relativity. Together they make up the geometric theory of differentiable manifolds - which can also be studied directly from the point of view of dynamical systems.

1 Differential geometry versus differential topology
2 Intrinsic versus extrinsic
3 Technical requirements
4 Differential topology
5 Branches of differential geometry
6 Applications of differential geometry
7 See also
8 External links
9 Notes
10 References

[edit] Differential geometry versus differential topology

For more details on this topic, see geometry and topology.

Differential topology and differential geometry are first characterized by their similarity. They both study primarily the properties of differentiable manifolds, sometimes with a variety of structures imposed on them.

One major difference lies in the nature of the problems that each subject tries to address. In one view,^[1] differential topology distinguishes itself from differential geometry by studying primarily those problems which are inherently global. Consider the example of a coffee cup and a donut (see this example.) From the point of view of differential topology, the donut and the coffee cup are the same (in a sense). A differential topologist imagines that the donut is made out of a rubber sheet, and that the rubber sheet can be smoothly reshaped from its original configuration as a donut into a new configuration in the shape of a coffee cup without tearing the sheet or gluing bits of it together. This is an inherently global view, though, because there is no way for the differential topologist to tell whether the two objects are the same (in this sense) by looking at just a tiny (local) piece of either of them. She must have access to each entire (global) object.

From the point of view of differential geometry, the coffee cup and the donut are different because it is impossible to rotate the coffee cup in such a way that its configuration matches that of the donut. This is also a global way of thinking about the problem. But an important distinction is that the geometer doesn't need the entire object to decide this. By looking, for instance, at just a tiny piece of the handle, she can decide that the coffee cup is different from the donut because the handle is thinner (or more curved) than any piece of the donut.

To put it succinctly, differential topology studies structures on manifolds which, in a sense, have no interesting local structure. Differential geometry studies structures on manifolds which do have an interesting local (or sometimes even infinitesimal) structure.

More mathematically, for example, the problem of constructing a diffeomorphism between two manifolds of the same dimension is inherently global since locally two such manifolds are always diffeomorphic. Likewise, the problem of computing a quantity on a manifold which is invariant under differentiable mappings is inherently global, since any local invariant will be trivial in the sense that it is already exhibited in the topology of Rⁿ. Moreover, differential topology does not restrict itself necessarily to the study of diffeomorphism. For example, symplectic topology — a subbranch of differential topology — studies global properties of symplectic manifolds. Differential geometry concerns itself with problems — which may be local or global — that always have some non-trivial local properties. Thus differential geometry may study differentiable manifolds equipped with a connection, a metric (which may be Riemannian, pseudo-Riemannian, or Finsler), a special sort of distribution (such as a CR structure), and so on.

This distinction between differential geometry and differential topology is blurred, however, in questions specifically pertaining to local diffeomorphism invariants such as the tangent space at a point. Differential topology also deals with questions like these, which specifically pertain to the properties of differentiable mappings on Rⁿ (for example the tangent bundle, jet bundles, the Whitney extension theorem, and so forth).

Nevertheless, the distinction becomes clearer in abstract terms. Differential topology is the study of the (infinitesimal, local, and global) properties of structures on manifolds having no non-trivial local moduli, whereas differential geometry is the study of the (infinitesimal, local, and global) properties of structures on manifolds having non-trivial local moduli.

[edit] Intrinsic versus extrinsic

Initially and up to the middle of the nineteenth century, differential geometry was studied from the extrinsic point of view: curves and surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions). The simplest results are those in the differential geometry of curves. Starting with the work of Riemann, the intrinsic point of view was developed, in which one cannot speak of moving 'outside' the geometric object because it is considered as given in a free-standing way.

The intrinsic point of view is more flexible. For example, it is useful in relativity where space-time cannot naturally be taken as extrinsic (what would be 'outside' it?). With the intrinsic point of view it is harder to define the central concept of curvature and other structures such as connections, so there is a price to pay.

These two points of view can be reconciled, i.e. the extrinsic geometry can be considered as a structure additional to the intrinsic one. (See the Nash embedding theorem.)

[edit] Technical requirements

The apparatus of differential geometry is that of calculus on manifolds: this includes the study of manifolds, tangent bundles, cotangent bundles, differential forms, exterior derivatives, integrals of p-forms over p-dimensional submanifolds and Stokes' theorem, wedge products, and Lie derivatives. These all relate to multivariable calculus; but for geometric applications, differential geometry must be developed in a way that makes good sense without a preferred coordinate system. The distinctive concepts of differential geometry can be said to be those that embody the geometric nature of the second derivative: the many aspects of curvature.

A real differentiable manifold is a topological space with a collection of diffeomorphisms from open sets of the space to open subsets in Rⁿ such that the open sets cover the space, and if f, g are diffeomorphisms then the composite mapping f o g ^-1 from an open subset of the open unit ball to the open unit ball is infinitely differentiable. We say a function from the manifold to R is infinitely differentiable if its composition with every diffeomorphism results in an infinitely differentiable function from the open unit ball to R. Of course manifolds need not be real, for example we can have complex manifolds.

At every point of the manifold, there is the tangent space at that point, which consists of every possible velocity (direction and magnitude) with which it is possible to travel away from this point. For an n-dimensional manifold, the tangent space at any point is an n-dimensional vector space, or in other words a copy of Rⁿ. The tangent space has many definitions. One definition of the tangent space is as the dual space to the linear space of all functions which are zero at that point, divided by the space of functions which are zero and have a first derivative of zero at that point. Having a zero derivative can be defined by "composition by every differentiable function to the reals has a zero derivative", so it is defined just by differentiability.

A vector field is a function from a manifold to the disjoint union of its tangent spaces (this union is itself a manifold known as the tangent bundle), such that at each point, the value is an element of the tangent space at that point. Such a mapping is called a section of a bundle. A vector field is differentiable if for every differentiable function, applying the vector field to the function at each point yields a differentiable function. Vector fields can be thought of as time-independent differential equations. A differentiable function from the reals to the manifold is a curve on the manifold. This defines a function from the reals to the tangent spaces: the velocity of the curve at each point it passes through. A curve will be said to be a solution of the vector field if, at every point, the velocity of the curve is equal to the vector field at that point.

An alternating k-dimensional linear form is an element of the antisymmetric k'th tensor power of the dual V^* of some vector space V. A differential k-form on a manifold is a choice, at each point of the manifold, of such an alternating k-form -- where V is the tangent space at that point. This will be called differentiable if whenever it operates on k differentiable vector fields, the result is a differentiable function from the manifold to the reals. A space form is a linear form with the dimensionality of the manifold.

[edit] Differential topology

Differential topology per se considers the properties and structures that require only a smooth structure on a manifold to define (such as those in the previous section). Smooth manifolds are 'softer' than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology. For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold—that is, one can smoothly "flatten out" certain manifolds, but it might require distorting the space and affecting the curvature or volume.

Conversely, smooth manifolds are more rigid than the topological manifolds. Certain topological manifolds have no smooth structures at all (see Donaldson's theorem) and others have more than one inequivalent smooth structure (such as exotic spheres). Some constructions of smooth manifold theory, such as the existence of tangent bundles, can be done in the topological setting with much more work, and others cannot.

[edit] Branches of differential geometry

[edit] Riemannian geometry

Riemannian geometry has Riemannian manifolds as the main object of study — smooth manifolds with additional structure which makes them look infinitesimally like Euclidean space. These allow one to generalise the notion from Euclidean geometry and analysis such as gradient of a function, divergence, length of curves and so on; without assumptions that the space is globally so symmetric. The Riemannian curvature tensor is an important pointwise invariant associated to a Riemannian manifold that measures how close it is to being flat.

[edit] Pseudo-Riemannian geometry

Pseudo-Riemannian geometry generalizes the flat Minkowski space of Einstein's special relativity to curved space. Pseudo-Riemannian geometry is the mathematical basis of Einstein's general relativity theory of gravity.

[edit] Finsler geometry

Finsler geometry has the Finsler manifold as the main object of study — this is a differential manifold with a Finsler metric, i.e. a Banach norm defined on each tangent space. A Finsler metric is a much more general structure than a Riemannian metric. A Finsler structure on a manifold M is a function F : TM → [0,n) such that:

F(x, my) = mF(x,y) for all x, y in TM,
F is infinitely differentiable in TM − {0},
The vertical Hessian of FF/2 is positive definite.

[edit] Symplectic geometry

Symplectic geometry is the study of symplectic manifolds. A symplectic manifold is a differentiable manifold equipped with a symplectic form <math>\omega </math> (that is, a non-degenerate, bilinear, skew-symmetric and closed 2-form). Since the symplectic form must be skew-symmetric, its matrix representation must be skew-symmetric, i.e.

It follows that <math> \det(M) = (-1)^n \det(M) </math> if <math> M </math> is an <math> n \times n </math> matrix. Thus, for odd <math>n</math> we see that <math> \det(M) = 0, </math> and so non-degenerate skew-symmetric two forms can only exist on even dimensional spaces. Unlike in Riemannian geometry, all symplectic manifolds are locally isomorphic: this is called Darboux's theorem and follows from the assumption that <math>\omega </math> is closed, so the only invariants of a symplectic manifold are global in nature. A diffeomorphism between two symplectic spaces which preserves the symplectic structure (i.e. the symplectic form) is called a symplectomorphism.

In dimension 2, a symplectic manifold is just a manifold endowed with an area form. The first result in symplectic topology is probably the Poincaré-Birkhoff theorem, conjectured by Henri Poincaré and proved by George David Birkhoff in 1912. This claims that if an area preserving map of a ring twists each boundary component in opposite directions, then the map has at least two fixed points.

It is easy to show that the area preserving condition (or the twisting condition) cannot be removed.

Note that if one tries to extend such a theorem to higher dimensions, one would probably guess that a volume preserving map of a certain type must have fixed points. This is false in dimensions greater than 3.

[edit] Contact geometry

Contact geometry is an analog of symplectic geometry which works for certain manifolds of odd dimension. Roughly, the contact structure on a (2n+1)-dimensional manifold is a choice of a hyperplane field that is nowhere integrable. This is equivalent to the hyperplane field being defined by a 1-form <math>\alpha</math> such that <math>\alpha\wedge (d\alpha)^n</math> does not vanish anywhere.

[edit] Complex and Kähler geometry

Complex differential geometry is the study of complex manifolds. An almost complex manifold is a real manifold <math>M</math>, endowed with a tensor of type (1,1), i.e. a vector bundle endomorphism (called an almost complex structure)

<math> J:TM\rightarrow TM </math>, such that <math>J^2=-1</math>.

It follows from this definition that an almost complex manifold is even dimensional.

An almost complex manifold is called complex if <math>N_J=0</math>, where <math>N_J</math> is a tensor of type (2,1) related to <math>J</math>, called the Nijenhuis tensor (or sometimes the torsion). An almost complex manifold is complex if and only if it admits a holomorphic coordinate atlas. An almost Hermitian structure is given by an almost complex structure J, along with a riemannian metric g, satisfying the compatibility condition <math>g(JX,JY)=g(X,Y)</math>. An almost hermitian structure defines naturally a differential 2-form <math>\omega_{J,g}(X,Y):=g(JX,Y)</math>. The following two conditions are equivalent:

<math> N_J=0\mbox{ and }d\omega=0,</math>
<math>\nabla J=0,</math>

where <math>\nabla</math> is the Levi-Civita connection of <math>g</math>. In this case, <math>(J, g)</math> is called a Kähler structure, and a Kähler manifold is a manifold endowed with a Kähler structure. In particular, a Kähler manifold is both a complex and a symplectic manifold. A large class of Kähler manifolds (the class of Hodge manifolds) is given by all the smooth complex projective varieties.

[edit] CR geometry

CR geometry is the study of the intrinsic geometry of boundaries of domains in complex manifolds.

[edit] Applications of differential geometry

Below are some examples of how differential geometry is applied to other fields of science and mathematics.

In physics, differential geometry is the language in which Einstein's general theory of relativity is expressed. According to the theory, the universe is a smooth manifold equipped with pseudo-Riemannian metric, which described the curvature of space-time. Understanding this curvature is essential for the positioning of satellites into orbit around the earth. Differential geometry is also indispensable in the study of gravitational lensing and black holes.
In economics, differential geometry has applications to the field of econometrics^[2].
Geometric modeling (including computer graphics) and computer-aided geometric design draw on ideas from differential geometry.
In engineering, differential geometry can be applied to solve problems in digital signal processing ^[3].
In physics, the use of differential forms is useful in the study of electromagnetism.
In physics, differential geometry has applications to both Lagrangian mechanics and Hamiltonian mechanics. Symplectic manifolds in particular can be used to study Hamiltonian systems.

[edit] See also

[edit] External links

[edit] Notes

^ Hirsch (1997)
^ Paul Marriott and Mark Salmon, "Applications of Differential Geometry to Econometrics".
^ Jonathan H. Manton, "On the role of differential geometry in signal processing" [1].

[edit] References

Theodore Frankel (2004). The geometry of physics: an introduction, second edition. ISBN 0-521-53927-7.
Spivak, Michael (1999). A Comprehensive Introduction to Differential Geometry (5 Volumes), 3rd Edition.
do Carmo, Manfredo (1976). Differential Geometry of Curves and Surfaces. ISBN 0-13-212589-7. A classical geometric approach to differential geometry without the tensor machinery.
do Carmo, Manfredo Perdigao (1994). Riemannian Geometry.
McCleary, John (1994). Geometry from a Differentiable Viewpoint.
Bloch, Ethan D. (1996). A First Course in Geometric Topology and Differential Geometry.
Gray, Alfred (1998). Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed..
Burke, William L. (1985). Applied Differential Geometry.
ter Haar Romeny, Bart M.. Front-End Vision and Multi-Scale Image Analysis. ISBN 1-4020-1507-0.
Hirsch, Morris (1997). Differential Topology. Springer-Verlag. ISBN 0-387-90148-5.